Geometry & Trigonometry: notes and practice questions

Distance between two points $P(x_1, y_1, z_1)$ and $Q(x_2, y_2, z_2)$ in 3D: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2}$.

Volume formulas: Cube: $V = a^3$.

Pythagorean Theorem is used only for right-angled triangles. It states that the square of the hypotenuse ($c$) equals the sum of the squares of the two shorter sides ($a$ and $b$): $a^2 + b^2 = c^2$. You must remember this formula as it is not provided in the booklet. Trigonometry (SOHCAHTOA) relates the ratios of side lengths to an angle ($\theta$) in a right-angled triangle.

Sine Rule: $\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}$. Cosine Rule: $c^2 = a^2 + b^2 - 2ab\cos C$.

Use Pythagoras' Theorem for right triangles: $a^2 + b^2 = c^2$.

Radians: $\theta$ (in radians) = $\frac{\text{arc length}}{\text{radius}}$. Arc Length: $l = r\theta$.

The unit circle has a radius of 1, centered at $(0,0)$, with trigonometric values derived from $(\cos \theta, \sin \theta)$. Exact values for angles $0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ$ are used frequently.

Pythagorean identity:. $\sin^2\theta + \cos^2\theta = 1$.

A periodic function repeats at regular intervals (e.g., sine and cosine functions). General form: $y = a \sin(bx + c) + d$ or $y = a \cos(bx + c) + d$:

Solve trig equations like $\sin x = a$, $\cos x = b$, $\tan x = c$ using exact values or GDC. Consider general solutions ($x + 360^\circ k$ or $x + 2\pi k$) and restrictions. For quadratic equations involving trig functions (e.g., $2\sin^2 x - \sin x - 1 = 0$), factorize or use the quadratic formula to solve for the trig ratio, then find angles.

Reciprocal trig functions (sec, csc, cot) are defined as reciprocals of cos, sin, and tan, respectively. They have related Pythagorean identities: $1 + \tan^2 x = \sec^2 x$ and $1 + \cot^2 x = \csc^2 x$. Solving equations involves converting to primary ratios. Inverse trig functions (arcsin, arccos, arctan) require restricting domains to ensure one-to-one correspondence. Each has a specific domain and principal range.

Pythagorean Identities: Fundamental relations like $\sin^2 \theta + \cos^2 \theta = 1$ simplify expressions. Reciprocal & Ratio Identities: Convert between trigonometric functions (e.g., $\tan \theta = \sin \theta / \cos \theta$).

Scalars have magnitude only; vectors have both magnitude and direction. Vectors can be represented as directed line segments or component forms (column or base vectors $\mathbf{i}, \mathbf{j}, \mathbf{k}$).

The scalar product (dot product) of two vectors yields a scalar value. It can be calculated algebraically ($v_1w_1 + v_2w_2 + v_3w_3$) or geometrically ($|\boldsymbol{v}||\boldsymbol{w}|\boldsymbol{\text{cos}}\theta$).

A line in 2D or 3D is defined by a fixed point and a direction vector. Vector equation: $r = a + \lambda b$, where $a$ is a position vector, $b$ is the direction vector, and $\lambda$ is a scalar parameter.

Lines in 2D can be coincident, parallel, or intersecting based on their slopes and y-intercepts. In 3D, lines are classified as parallel, coincident, intersecting, or skew.

The vector product (cross product) is defined only for 3D vectors, resulting in a vector perpendicular to both operands. It can be calculated algebraically using a determinant formula or geometrically via magnitude $|\boldsymbol{a} \times \boldsymbol{b}| = |\boldsymbol{a}| |\boldsymbol{b}| \boldsymbol{\boldsymbol{\text{sin}}} \theta$.

A plane can be represented by a vector equation ($r = a + \lambda u + \mu v$), parametric equations, or a scalar product form ($r \cdot n = a \cdot n$). The Cartesian equation of a plane is $ax + by + cz = d$, where $n = \begin{pmatrix} a \\ b \\ c \end{pmatrix}$ is the normal vector.

Lines can be represented in vector, parametric, or Cartesian form, using a position vector and a direction vector. Planes can be represented in vector, scalar product, or Cartesian form.